YES Termination w.r.t. Q proof of ../tpdb/TRS/Rubio/bintrees.trs
Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

Q restricted rewrite system:
The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Q is empty.

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

concat(leaf, Y) → Y
concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

concat(leaf, Y) → Y
lessleaves(X, leaf) → false
lessleaves(leaf, cons(W, Z)) → true
Used ordering:
Polynomial interpretation [25]:

POL(concat(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(false) = 2   
POL(leaf) = 2   
POL(lessleaves(x1, x2)) = 2·x1 + 2·x2   
POL(true) = 2   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

Q restricted rewrite system:
The TRS R consists of the following rules:

concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Q is empty.

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

concat(cons(U, V), Y) → cons(U, concat(V, Y))
lessleaves(cons(U, V), cons(W, Z)) → lessleaves(concat(U, V), concat(W, Z))
Used ordering:
Polynomial interpretation [25]:

POL(concat(x1, x2)) = 1 + 2·x1 + x2   
POL(cons(x1, x2)) = 2 + 2·x1 + x2   
POL(lessleaves(x1, x2)) = 2·x1 + x2   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RisEmptyProof

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.